Has anyone ever heard the phrase "taking him going away" used to describe the experience of hooking a fish ("him") while it is moving away from/in the opposite direction of the person fishing?
This looks to be more of a hunting phrase from these two references I found:
I'm not familiar with this being adopted into fishing lingo—at least not in fly-fishing. Although there may be similar terms for the different ways a fish can take a fly. If a fish is hit by a hook and caught other than by its mouth, it is said to be snagged.
I've not heard it exactly like that but I fully understand the meaning. When a fish takes bait there is a window of opportunity to strike to ensure a secure contact between hook and fish. That window of opportunity varies between fish and bait (and many other factors).
"Taking him going away", I would interpret as let the fish run and then strike. Another similar example would be - Let the carp head-off and strike when the float is under